Orbits of real forms in complex flag manifolds

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Tesi di Dottorato

Orbits of real forms

in complex flag manifolds

Andrea Altomani Dottorando XX Ciclo

Universit`a di Roma “Tor Vergata”

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Introduction

A complex flag manifold is a compact complex manifold M that is homogeneous for the action of a semisimple complex Lie group ˆG; equivalently M is of the form M= ˆG/Q, with Q a complex parabolic subgroup of ˆG. The orbits M in M of a real form G of ˆG inherit from the complex structure of M a G-homogeneous CR structure. In this way we obtain a large class of CR manifolds, that we call parabolic CR manifolds. They are homogeneous for the CR action of a real semisimple Lie group. Special examples are the compact standard homogeneous CR manifolds, corresponding to Levi-Tanaka algebras (see e.g. [MN97], [MN98], [MN01]), and the symmetric CR manifolds in [LN05].

The orbits of G in M were already considered in [Wol69]. Here it is proved that there is a unique compact G-orbit in M, that we call compact parabolic CR manifold.

Among the several recent contributions to the study of this subject, we cite [Kas93] in the context of infinite dimensional representation theory, [GM03], [HW03] and [KZ03] for applications to the geometry of symmetric spaces.

In this work we stress the point of view of CR geometry. The main tool we use are parabolic CR algebras, that is CR algebras of the form (g, q), where g is a real semisimple Lie algebra and q is a parabolic subalgebra of the complexification ˆ

g _{= C ⊗ g of g. These algebras, first introduced in [MN05], provide an algebraic} description of the local CR structure of homogeneous CR manifolds.

It is possible to find Cartan subalgebras h of g contained in q. Several CR and topological invariants of M can thus be described in terms of carefully chosen bases of the root system R(ˆg, ˆh) of ˆg with respect to ˆh_{= C ⊗ h.}

The open orbits are complex manifolds and have been extensively studied (see e.g. [FHW06]). In particular, they are all simply connected (see [Wol69]). Also the topology of the real flag manifolds has been thoroughly investigated (see e.g. [CS99], [DKV83], [Wig98]). In this work we show that every parabolic CR manifold M is the total space of a canonical fibration over a real flag manifold. The fiber may be disconnected and each connected component is a simply connected complex manifold, which can be retracted onto an open orbit. This essentially reduces the computation of the fundamental group of M to counting the connected components of the fiber.

The thesis is organized as follows.

The first part deals with general parabolic CR manifolds and comprises Chap-ters 1–4.

In Chapter 1 we review the notions of CR algebras and homogeneous CR man-ifolds from [MN05], that was also recently utilized in [Fel06] and [FK06]. We collect here the main results and fix the notation that will be employed in the following chapters.

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In Chapter 2 we quickly rehearse parabolic complex Lie subalgebras and com-plex flag manifolds and begin the study of the CR algebras that are associated to the real orbits M in the complex flag manifolds M, also investigating the canonical G-equivariant maps of [MN05] in this special situation.

Chapter 3 is the core of our investigation of the CR properties of M . Through the introduction of adapted Cartan subalgebras and S- and V-fit Weyl chambers, we associate to M special systems of simple roots. Weak (i.e. holomorphic accord-ing to [BER99]) nondegeneracy and fundamentality (i.e. finite type accordaccord-ing to [BG77]) are proved to be equivalent to properties of these systems of simple roots. These can be checked from the pattern of some cross marked diagrams associated to M , that generalize those of [MN98], [LN05].

In Chapter 4 we turn to the construction of homogeneous CR manifolds that fiber over our orbit M and that are useful both for finding the S- and V-fit Weyl chambers and for investigating the topological properties of M in the following sections. In particular, we construct the weakest CR model of M , that is a step to build a chain of fibrations, with simply connected fibers, that in some instances coincides with, and in general can be considered as a substitute of, the holomorphic arc components of [Wol69].

In the second part, that comprises Chapters 5–7, the special case of compact parabolic CR manifolds is studied in detail.

In Chapter 5 we characterize those parabolic CR algebras that correspond to compact CR manifolds and associate to them a special subclass of the diagrams introduced in Chapter 3. Then we study G-equivariant fibrations of compact par-abolic CR manifolds and classify totally real and totally complex ones.

In Chapter 6 we investigate several nondegeneracy conditions for compact par-abolic CR manifolds, sharpening the results of Chapter 3.

In Chapter 7 we recall the definition of essential pseudoconcavity, a notion that generalizes that of pseudoconcavity, and characterize compact parabolic CR manifolds that are essentially pseudoconcave.

The third part of the thesis, that includes Chapters 8–10, presents some appli-cations of the theory developed in the previous chapters.

In Chapter 8 we investigate the connectivity of the isotropy subgroup of M . This is needed to study the connectivity of the fibers of a fibration of M over a real flag manifold M0, that we utilize to compute the fundamental group of M . This is a somehow delicate point: the simply connected fibers of our construction may be not connected. We use classical results from [BT72], [BT65] to characterize Cartan subgroups and isotropy subgroups of connected semisimple real linear groups in terms of characters. Then we discuss the fundamental group of M .

In Chapter 9 we provide several examples which show how effective our results are for the study of CR and topological properties of the orbits.

Finally, in Chapter 10, we describe the space of global CR functions on para-bolic CR manifolds

All the results contained in this thesis were first presented in [AM06], [AMN06a], [AMN06b], [Alt07].

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Part 1

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CHAPTER 1

Homogeneous CR manifolds and CR algebras

In this chapter we review some aspects of the theory of homogeneous CR mani-folds. First we recall the basic definitions and results about CR manifolds and CR maps (a general reference for this topic, covering much more than we need here, is [BER99]). Then we introduce homogeneous CR manifolds and review the relation between (germs of) homogeneous CR manifolds and CR algebras, along the lines of [MN05].

1.1 CR manifolds

A CR manifold of type (n, k) is a triple (M, HM, J ), consisting of: (1) a smooth paracompact manifold M of real dimension (2n + k),

(2) a smooth real vector subbundle HM of rank 2n of its real tangent bundle T M ,

(3) a smooth complex structure J : HM −→ HM on the fibers of HM .

The integers n and k are the CR dimension and CR codimension of M . It is also required that J satisfies the formal integrability conditions :

(1.1) C∞(M, T0,1M ), C∞(M, T0,1M ) ⊂ C∞(M, T0,1M ).

where T0,1M = {X + iJ X | X ∈ HM } is the complex subbundle of the complexifi-cation CHM of HM corresponding to the eigenvalue −i of J; with T1,0M = T0,1_M

we have T1,0M ∩ T0,1M = 0 and T1,0M ⊕ T0,1_{M = CHM . Any real smooth} man-ifold is a CR manman-ifold, with n = 0. When k = 0 instead, we recover the abstract definition of a complex manifold, via the Newlander-Nirenberg theorem.

If (M, HM, J ) and (M0, HM0, J0) are CR manifolds, a smooth f : M −→ M0 _is

a CR map if :

(1) df (HM ) ⊂ HM0,

(2) df ◦ J = J0◦ df on HM .

Assume that (M0, HM0, J0) is a CR manifold and f : M −→ M0 _{a smooth immersion.}

For x ∈ M we define HxM and Jxv for v ∈ HxM by setting :

(1.2)

_H

xM = [df (x)]−1 [df (TxM ) ∩ Hf (x)M0] ∩ [J0(df (TxM ) ∩ Hf (x)M0)]

Jx(v) = [df (x)]−1(J0([df (x)](v))) .

If the dimension HxM is a constant integer, independent of x ∈ M , then the

dis-joint union HM of the HxM ’s, and the map J : HM −→ HM , equal to Jx on

the fiber HxM , define a CR manifold (M, HM, J ). This is the CR structure on

M with the maximal CR dimension among those for which f is a CR map. In this case the map f : M −→ M0 _{is called a CR immersion. If (M}0_{, HM}0_{, J}0_{) is of}

type (n0, k0) and (M, HM, J ) of type (n, k), we always have n + k ≤ n0+ k0. The 3

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immersion is generic when the equality n + k = n0+ k0 holds. When f : M −→ M0

is also an embedding, we say that f is a CR embedding or a generic CR embedding, respectively.

A CR map f : (M, HM, J ) −→ (M0, HM0, J0) is a CR submersion if f : M −→ M0 is a smooth submersion and moreover df (x)(HxM ) = Hf (x)M0 for all x ∈ M . If

(M, HM, J ) is of type (n, k) and (M0, HM0, J0) of type (n0, k0), the existence of a CR submersion implies that n ≥ n0 and k ≥ k0.

When f : M −→ M0 is a CR submersion and a smooth fiber bundle, we say that f : (M, HM, J ) −→ (M0_{, HM}0_{, J}0_{) is a CR fibration. The fibers are embedded CR}

submanifolds of M0 of type (n − n0, k − k0).

A CR diffeomorphism of (M, HM, J ) onto (M0, HM0, J0) is a diffeomorphism f : M −→ M0 _{such that both f and f}−1 _{are smooth CR maps. The set of all CR}

diffeomorphisms of (M, HM, J ) onto itself (CR automorphisms) is a group with the composition operation.

We say that (M, HM, J ) is a homogeneous CR manifold if there is a Lie group of CR automorphisms that acts transitively on M .

Let (M, HM, J ) be a CR manifold. A vector field X ∈ C∞(U, T M ), defined on an open subset U of M , is an infinitesimal CR automorphism if the maps ϕX(t)

of the local 1-parameter group of local transformations generated by X are CR. This is equivalent to the fact that [X, C∞(U, T0,1M )] ⊂ C∞(U, T0,1M ). We say that (M, HM, J ; o) is a locally homogeneous CR manifolds at a point o ∈ M if, for each v ∈ ToM , there is an infinitesimal CR automorphism X, defined in an open

neighborhood U of o in M , with v = X(o).

Homogeneous CR manifolds are locally CR homogeneous: a homogeneous CR manifold (M, HM, J ) has a real analytic CR structure and therefore (see e.g. [AF79]) admits a generic embedding M ,→ ˆM into a complex manifold ˆM . Then the Lie algebra g of the Lie group G that acts transitively by CR automorphisms on M can be identified with a Lie algebra of infinitesimal analytic CR automor-phisms defined on U = M . Each X∗ ∈ C∞(M, T M ), corresponding to an X ∈ g, is the restriction of the real part of a holomorphic vector field Z∗, defined on an open complex neighborhood ˆU of M in ˆM (i.e. X∗ = [Re Z∗] |M; see e.g. [BER99,

§12.4]).

The germs of infinitesimal CR automorphisms of (M, HM, J ) at a point o ∈ M , with the Lie bracket, form a real Lie algebra G = G(M, HM, J ; o). We consider its complexification ˆG _{= C ⊗ G and denote by Q = Q(M, HM, J; o) the complex} Lie subalgebra of ˆG consisting of all Z ∈ ˆGwith Z(o) ∈ T_o0,1M . The fact that Q is actually a complex Lie subalgebra of ˆG is a consequence of the formal integrability of the partial complex structure J .

When (M, HM, J ) is locally CR homogeneous at o ∈ M , the pair (G, Q) = (G(M, HM, J ; o), Q(M, HM, J ; o)) completely determines the germ of the CR manifold (M, HM, J ) at o. Vice versa, if g is a finite dimensional real Lie algebra and q a complex Lie subalgebra of its complexification ˆg, the general construction1

of a germ (M, o) of homogeneous manifold associated to the Lie algebra g and to its

1_{If g is a finite dimensional real Lie algebra and g}

+ a real Lie subalgebra of g, we can find a

germ (M, o) of analytic real manifold, unique modulo germs of analytic diffeomorphisms, for which there is a real Lie algebras homomorphism ı : g −→ C∞

(o)(M, T M ) with {ı(X)(o) | X ∈ g} = ToM

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1.1. CR MANIFOLDS 5

Lie subalgebra g+ = g ∩ q (cf. e.g. [Pos86, Ch.VII]) yields a unique, modulo local

CR diffeomorphisms, germ of locally homogeneous CR manifold (M, HM, J ; o) at o ∈ M , such that the complexification ˆı of the correspondence ı : g −→ C∞

(o)(M, T M )

yields a homomorphism of complex Lie algebras: ˆı : ˆg−→ ˆG(M, HM, J ; o) with (1.3) ı(g) ⊂ G(M, HM, J ; o) , ˆı(q) ⊂ Q(M, HM, J ; o)

and for which the induced map on the quotients

g/(g ∩ q) −→ G(M, HM, J ; o)/ (G(M, HM, J ; o) ∩ Q(M, HM, J ; o))

is an isomorphism. In this case we say that the germ of (M, HM, J ) at o is associ-ated to the pair (g, q). (We shall consistently use “hat ” to indicate complexification: e.g. ˆϕ : ˆg−→ ˆg0 is the complexification of ϕ : g −→ g0).

These remarks led to the introduction in [MN05] of the abstract notion of a CR algebra. A CR algebra (g, q) is the pair consisting of a real Lie algebra g and of a complex Lie subalgebra q of its complexification ˆg_{= C ⊗ g, such that the quotient} g/ (g ∩ q) is a finite dimensional real linear space. Note that we do not require that g is finite dimensional. The intersection g+ = g ∩ q is the isotropy of (g, q). Let

H+ = {Re Z | Z ∈ q} and denote by ¯Z the conjugate of Z ∈ ˆg with respect to the

real form g.

A CR algebra (g, q) is :

• totally real if H+ = g+,

• totally complex if H+ = g,

• fundamental if H+ generates g as a real Lie algebra,

• transitive, or effective if g+ does not contain any nonzero ideal of g,

• ideal nondegenerate if all ideals of g contained in H+ are contained in

g+,

• weakly nondegenerate if there is no complex Lie subalgebra q0 of ˆgwith : q ( q0 ⊂ q + ¯q,

• strictly nondegenerate if g+ = {X ∈ H+| [X, H+] ⊂ H+}.

Clearly :

strictly nondegenerate =⇒ weakly nondegenerate =⇒ ideal nondegenerate . Fundamentality of (g, q) is equivalent to the fact that the associated germ of homogeneous CR manifold (M, HM, J ; o) is of finite type in the sense of [BG77], i.e. that the smallest involutive distribution of tangent vectors containing HM also contains ToM .

Strict and weak nondegeneracy hold, or do not hold, for all CR algebras that are associated to the same germ (M, HM, J ; o) of locally homogeneous CR manifold. They correspond indeed to the nondegeneracy of the (vector valued) Levi form and of its higher order analog, respectively (see e.g. [MN05, §13]). In particular, weak nondegeneracy at a point o ∈ M of a (germ of) CR man-ifold (M, HM, J ) means that, for every L ∈ C∞(M, T1,0M ) with L(o) 6= 0, there exist finitely many vector fields ¯Z1, . . . , ¯Zm ∈ C∞(M, T0,1M ) such that

[L , ¯Z1, . . . , ¯Zm](o) /∈ To1,0M ⊕ To0,1M .

When (g, q) defines at o a germ of homogeneous CR manifold, the two notions of weak nondegeneracy, the one for CR algebras and the one above for CR man-ifolds, coincide and also coincide with the holomorphic nondegeneracy of [BER99] and the finite nondegeneracy of [Fel06].

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Proposition 1.1. Let (M, HM, J ) and (M0, HM0, J0) be CR manifolds. Assume that M0 is locally embeddable and that there exists a CR fibration π : (M, HM, J ) −→ (M0_{, HM}0_{, J}0_{), with totally complex fibers of positive}

dimen-sion. Then M is weakly degenerate.

Proof. Let f be any smooth CR function defined on a neighborhood U0 of p0 ∈ M0. Then π∗f is a CR function in U = π−1(U0), that is constant along the fibers of π. Hence, if L ∈ C∞(M, T1,0_{M ) is tangent to the fibers of π in}

U , we obtain that [ ¯Z1, . . . , ¯Zm, L] (π∗f ) = 0 for every choice of ¯Z1, . . . , ¯Zm ∈

C∞_{(M, T}0,1_{M ). Assume by contradiction that M is weakly nondegenerate at some}

p with π(p) = p0. Then for some choice of ¯Z1, . . . , ¯Zm ∈ C∞(M, T0,1M ) we would

have vp = [ ¯Z1, . . . , ¯Zm, L] /∈ Tp1,0M ⊕ Tp0,1M . Since the fibers of π are totally

complex, π∗(vp) 6= 0. By the assumption that M0 is locally embeddable at p,

the real parts of the (locally defined) CR functions give local coordinates in M0 and therefore there is a CR function f defined on a neighborhood U0 of p0 with vp(π∗f ) = π∗(vp)(f ) 6= 0. This gives a contradiction, proving our statement.

Differently, both ideal degenerate and ideal nondegenerate CR algebras may correspond to the same (weakly degenerate) germ of locally homogeneous CR man-ifold.

From [MN05, Theorem 9.1] we know that if (g, q) is fundamental, effective, and ideal nondegenerate, then g is finite dimensional.

From this result we deduce the following :

Theorem 1.2. Let (g, q) be a fundamental effective CR algebra. Then there exist a germ of homogeneous complex manifold ( ˆM , o) at a point o, and a germ of homogeneous generic CR submanifold (M, HM, J ; o) of ( ˆM , o) at o, with associ-ated CR algebra (g, q).

Proof. First we note that the statement holds true when g is finite dimen-sional: by [Pos86, Ch.VII] there is a germ of homogeneous complex manifold ( ˆM , o) at o corresponding to the complex Lie algebra ˆg and to its complex Lie subalgebra q; the inclusion g ,→ ˆg yields the embedding into ( ˆM , o) of a germ of homogeneous CR manifold (M, HM, J ; o) at o, corresponding to the pair (g, q).

Consider now the general case. We keep the notation introduced above. By [MN05, Lemma 7.2] there is a largest ideal a of g contained in H+. By [MN05,

Theorem 9.1], g/a is finite dimensional and by the first part of the proof there is a germ of complex homogeneous manifold ( ˆN , o) at o, and a germ of generic CR submanifold (N, HN, JN; o) of ˆN at o, associated to the pair (g/a, q/(q ∩ ˆa)) (here

ˆ

a_{= C⊗}_Rais the complexification of a in ˆg). If 2d is the real dimension of a/(a∩g+),

we can take ˆM = ˆ_{N × C}d _{and, likewise, M = N × C}d_{, with HM = HN × T (C}d) and J = JN × J_Cd. Then (M, HM, J ; (o, 0)) is associated to (g, q). Note that the ideal nondegeneracy of (g, q) implies that all ideals x of the com-plex Lie algebra ˆg that are contained in q are contained in q ∩ ¯q. Indeed, if x is a (complex) ideal of ˆg contained in q, then a = (x + ¯x) ∩ g is an ideal of g contained in H+ = (q + ¯q) ∩ g, and a ⊂ g+= q ∩ g implies that x + ¯x⊂ q ∩ ¯q.

Let (g, q) be a CR algebra with a finite dimensional g. We denote by ˆG the connected and simply connected complex Lie group with Lie algebra ˆgand by Q its

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1.1. CR MANIFOLDS 7

analytic Lie subgroup, generated by q. Let G be the analytic subgroup of ˆG with Lie algebra g and set G+ = G+(g, q) := Q ∩ G. This is a Lie subgroup of G with

Lie subalgebra g+. Denote by ˜G a connected and simply connected Lie group with

Lie algebra g and by ˜G+ = ˜G+(g, q) its analytic Lie subgroup with Lie subalgebra

g+. Since ˆG is simply connected, the conjugation σ of ˆg with respect its real form

g defines an antiholomorphic involution, still denoted by σ, in ˆG. Thus G, being the connected component of the identity in the set of fixed points of σ, is closed in

ˆ

G. We have the implications :

Q closed in ˆG =⇒ G+ closed in G , G+ closed in G =⇒ ˜G+ closed in ˜G .

When ˜G+ is closed in ˜G, we can uniquely define a ˜G-homogeneous CR

mani-fold ˜M (g, q) = ( ˜M , H ˜M , ˜J ), where the underlying smooth manifold ˜M is the ˜ G-homogeneous space ˜G/ ˜G+, and (g, q) is associated to the germ ( ˜M , H ˜M , ˜J ; o) at

the base point o = e ˜G+.

Likewise, for a closed G+ ⊂ G, we define the G-homogeneous CR manifold

M (g, q) = (M, HM, J ) with M = G/G+ and (g, q) associated to (M, HM, J ; o)

for o = e G+.

If Q is closed, ˆM = ˆM (ˆg, q) := ˆG/Q is a ˆG-homogeneous complex manifold and M (g, q) can be identified, its partial complex structure being that of a generic CR submanifold of ˆM , to the orbit of G through the base point o = e Q of ˆM .

Our canonical choice of M (g, q) aims to obtain a homogeneous CR manifold with a generic CR embedding into a homogeneous complex manifold ˆM = ˆM (ˆg, q), that is “good” in some suitable sense.

A morphism of CR algebras (g, q) −→ (gϕ 0_{, q}0_{) is a homomorphism of real Lie}

algebras ϕ : g −→ g0, with ˆϕ(q) ⊂ q0. It is called :

• a CR immersion if the quotient map [ϕ] : g/g+ −→ g0/g0+ is injective and

ˆ

ϕ−1(q0) = q;

• a CR submersion if both [ϕ] : g/g+ −→ g0/g0+ and [ ˆϕ] : q/ˆg+ −→ q0/ˆg0+ are

onto;

• a local CR isomorphism if both [ϕ] : g/g+ −→ g0/g0+and [ ˆϕ] : q/ˆg+ −→ q0/ˆg0+

are isomorphisms;

• a CR isomorphism if ϕ is an isomorphism of real Lie algebras with ˆ

ϕ(q) = q0. We quote from [MN05] :

Proposition 1.3. Let (g, q) −→ (gϕ 0, q0) be a morphism of CR algebras, with g and g0 finite dimensional. Let (M, HM, J ; o) and (M0, HM0, J0; o0) be the germs of homogeneous CR manifolds at o ∈ M , o0 ∈ M0, associated to (g, q), (g0, q0), respectively. Then there is a unique germ of smooth CR map Φ : (M, HM, J ; o) −→ (M0, HM0, J0; o0) with Φ(o) = o0 such that dΦo(ı(X)) =

ı0(ϕ(X)). Here ı, ı0 are the homomorphisms of Lie algebras ı : g −→ G(M, HM, J ; o) and ı0 : g :−→ G(M0, HM0, J0; o) of (1.3).

The germ Φ of smooth CR map is a CR immersion, submersion, diffeomorphism if and only if the corresponding morphism ϕ of CR algebras is a CR immersion, a CR submersion, a local CR isomorphism, respectively.

Let ˜G and ˜G0 be the connected and simply connected real Lie groups with Lie algebras g and g0, respectively. If the analytic subgroup ˜G+ of ˜G with Lie algebra

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g+ = q ∩ g and the analytic subgroup ˜G0+ of ˜G0 with Lie algebra g0+ = q0 ∩ g0 are

both closed, then there is a unique smooth CR map ˜Φ : ˜M (g, q) −→ ˜M (g0, q0) that makes the following diagram commute :

g −−−−→ ˜exp G −−−−→ M (g, q)˜ ϕ   y   y   yΦ˜ g0 −−−−→ ˜exp G0 −−−−→ ˜M (g0, q0)

The map ˜Φ is a CR immersion, a CR submersion or a local CR diffeomorphism if and only if the corresponding CR morphism of CR algebras ϕ is a CR immersion, a CR submersion or a local CR isomorphism, respectively.

Let ˆG and ˆG0 be the connected and simply connected complex Lie groups with Lie algebras ˆg and ˆg0, respectively. Let G, Q ⊂ ˆG and G0, Q0 ⊂ ˆG0 be the analytic subgroups with Lie algebras g, q and g0, q0, respectively. If G+= Q ∩ G and G0 =

Q0∩ G0 _{are closed, then there is a unique smooth CR map Φ : M (g, q) −}_{→ M (g}0_{, q}0₎

such that the diagram :

˜ M (g, q) −−−−→ ˜Φ˜ M (g0, q0)   y   y M (g, q) −−−−→ M (gΦ 0_{, q}0₎

where the vertical arrows are the natural projections from the universal coverings, commutes.

The map Φ is a CR immersion, a CR submersion or a local CR diffeomorphism if and only if the corresponding morphism of CR algebras ϕ is a CR immersion, a CR submersion or a local CR isomorphism, respectively.

If Q ⊂ ˆG and Q0 ⊂ ˆG0 are closed, the map M (g, q) −→ M (gΦ 0_{, q}0_{) is the}

re-striction of the holomorphic map ˆΦ : ˆM = ˆG/Q −→ ˆM0 = ˆG0/Q0 defined by the commutative diagram : ˆ g −−−−→ ˆexp G −−−−→ Mˆ ˆ ϕ   y   y   yΦˆ ˆ g0 −−−−→ ˆexp G0 −−−−→ ˆM0

where the central vertical arrow is the homomorphism of complex connected sim-ply connected Lie algebras defined by the homomorphism ˆϕ : ˆg −→ ˆg0 of their Lie

algebras.

To discuss, later on, the structure of the fibers of some CR fibrations, we need to introduce the notion of semidirect sum of CR algebras.

Let (g1, q1), (g2, q2) be CR algebras, and assume that g2 has a g1-module

struc-ture and that q2 is a q1-module for the restriction of the complexification of the

action of g1 on g2. Then q = q1o q2 (semidirect sum) is a complex Lie subalgebra

of the complexification of the semidirect sum g = g1 o g2, and the CR algebra

(g, q) = (g1 o g2, q1o q2) is called the semidirect sum of the CR algebras (g1, q1)

and (g2, q2) :

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1.2. g-EQUIVARIANT FIBRATIONS 9

We shall assume that g1 and g2 are finite dimensional. Denote by :

ˆ

G, ˆG1, ˆG2 the connected and simply connected complex Lie groups with Lie

algebras ˆg, ˆg1, ˆg2, respectively;

G, G1, G2the analytic real subgroups of the corresponding complex connected

Lie groups ˆG, ˆG1, ˆG2, with Lie algebras g, g1, g2, respectively;

Q ⊂ ˆG, Q1 ⊂ ˆG1, Q2 ⊂ ˆG2 the Lie subgroups corresponding to the Lie

subalgebras q ⊂ ˆg, q1 ⊂ g1, q2 ⊂ g2, respectively;

˜

G, ˜G1, ˜G2 connected and simply connected real Lie groups with Lie algebras

g, g1, g2, respectively;

G+ = Q ∩ G, G1 + = Q ∩ G1, G2 + = Q ∩ G2;

˜

G+ ⊂ ˜G, ˜G1 + ⊂ ˜G1, ˜G2 + ⊂ ˜G2 the analytic subgroups corresponding to the

Lie subalgebras g+ = q ∩ g, g1+ = q1∩ g1, g2+ = q2 ∩ g2, respectively.

Let (M, HM, J ; o), (M1, HM1, J1; o1),(M2, HM2, J2; o2) be the germs of

lo-cally homogeneous CR manifolds associated to the CR algebras (g, q), (g1, q1),

(g2, q2), respectively.

We obtain :

Theorem 1.4. The diffeomorphism G1 × G2 3 (g1, g2) −→ g1g2 ∈ G1 o G2

defines a germ of CR diffeomorphism :

(M1, HM1, J1; o1) × (M2, HM2, J2; o2) −→ (M, HM, J ; o) .

If ˜G1 + and ˜G2 + are closed, then ˜G+ is closed and we obtain a global CR

diffeo-morphism :

˜

M (g1, q1) × ˜M (g2, q2) −→ ˜M (g, q) .

If G1 + and G2 + are closed, then G+ is closed and we obtain a global CR

diffeo-morphism :

(1.5) M (g1, q1) × M (g2, q2) −→ M (g, q) .

When Q1 and Q2 are closed, also Q is closed and the map (1.5) is the restriction

of a biholomorphic map ˆ_G₁_/Q₁ × ˆG2/Q2 −→ ˆG/Q . 1.2 g-equivariant fibrations

Let g be a real Lie algebra and q, q0 complex subalgebras of its complexification ˆ

g, with q ⊂ q0. Then the identity map in g and the inclusion q ,→ q0 define a g-equivariant morphism of CR algebras

(1.6) (g, q) −→ (g, q0) .

If (M, HM, J ; o) and (M0, HM0, J0; o0) are germs of locally homogeneous CR manifolds with associated CR algebras (g, q) and (g, q0), respectively, then the identity map g −→ g defines, by passing to the quotients, the differential of a CR map π(o) : (M, HM, J ; o) −→ (M0, HM0, J0; o0) that is locally G-equivariant for a

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Let G be a connected Lie group with Lie algebra g and assume that there are two G-homogeneous CR manifolds (M, HM, J ), (M0, HM0, J0) that are associated to (g, q) at some o ∈ M and to (g, q0) at some o0 ∈ M0_{, respectively. Then there is}

a unique G-equivariant CR map π : (M, HM, J ) −→ (M0, HM0, J0) with π(o) = o0. In general π(o) (and π, when defined) are smooth, but not CR, G-equivariant

fibrations: a necessary and sufficient condition for (1.6) to be a g-equivariant CR fibration, and hence for π(o) (and π, when defined) to be G-equivariant local (resp.

global) CR fibrations is that (see [MN05, Lemma 5.1])

(1.7) q0 = q + ˆg0₊.

We call (g, q0) the basis of the fibration (1.6). The fiber of (1.6) is the CR al-gebra (g0₊, q00) where q00 = ˆg0₊ ∩ q. It is a CR algebra associated to the germ (F, HF, J |HF ; o), where (F ; o) =

π−1_(o)(o0); o, and the germ of partial complex structure (HF, J |HF ; o) that is characterized by requiring that the smooth

em-bedding (π−1(o0); o) ,→ (M ; o) is a CR immersion.

We know that (1.6) is always a CR fibration, with a totally complex fiber, when q⊂ q0 _{⊂ q + q : indeed in this case q}0 _{= q + ˆ}_g0

+.

From [MN05, §5] we have :

Proposition 1.5. Let (g, q) be a CR algebra. Then there exist : • a largest ideal i of g with i ⊂ g+;

• a largest ideal a of g with a ⊂ H+;

• a largest complex Lie subalgebra q0 of ˆg with q ⊂ q0 ⊂ q + ¯q; • a smallest complex Lie subalgebra q00 _{of ˆ}_g _{with q + ¯}_q_{⊂ q}00_.

We have i ⊂ a ⊂ q0 ⊂ q00 and q00 = ¯q00 = ˆg00 for a real Lie subalgebra g00 of g. The identity in g defines g-equivariant CR fibrations (g, q) −→ (g, q0_{) −}_{→ (g, q}00_),

where (g, q0) is weakly nondegenerate and (g, q00) is totally real. For all complex Lie subalgebras f of ˆg _{with q ⊂ f ( q}0, the CR algebra (g, f) is weakly degenerate. For all complex Lie subalgebras f with q ⊂ f ⊂ q0, the g-equivariant map (g, q) −→ (g, f) is a CR fibration with a totally complex fiber.

The CR algebra (g00, q) is fundamental and, for all real Lie subalgebras l of g with g00 _{( l ⊂ g the CR algebra (l, q) is not fundamental.}

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CHAPTER 2

Parabolic CR algebras and

parabolic CR manifolds

In the first section of this chapter we collect the notions on complex parabolic sub-algebras and fix the notation that will then be utilized throughout this work. This is mostly a review of classical results, for which general references are [Bou02, Ch.IV §2.6, Ch.VI §1], [Bou05, Ch.VIII §3], [Kna02, Ch.VII], [War72, Ch.1], [Wol69].

In the second section we introduce the main object of our study, namely par-abolic CR algebras and parpar-abolic CR manifolds, and begin to study some of their properties.

2.1 Parabolic subalgebras and complex flag manifolds

Let ˆg be a complex Lie algebra. A maximal solvable complex Lie subalgebra b of ˆg is called a Borel, or minimal parabolic complex Lie subalgebra of ˆg. A complex Lie subalgebra q of ˆg is parabolic if it contains a complex Borel subalgebra b of ˆg.

For our purposes, it will be sufficient to consider the case of a semisimple ˆg. Thus from now on we shall assume that ˆg is a semisimple complex Lie algebra.

A parabolic subalgebra q of ˆg contains a complex Cartan subalgebra ˆh of ˆg. Let R = R(ˆg, ˆh) be the root system of ˆg with respect to ˆh. We denote by h_R the real form of ˆh on which all roots are real valued. Thus R is a subset of the real dual space h∗

R. The Killing form κˆg of ˆg restricts to a real positive scalar product

in h_R. We shall write (A|B) = κˆg(A, B) for A, B ∈ hR. We set also (ξ|η) = (Tξ|Tη)

for ξ, η ∈ h∗

R and (Tξ|A) = ξ(A), (Tη|A) = η(A) for all A ∈ hR (dual scalar product

in h∗_R). Roots α, β ∈ R for which α ± β /∈ R are called strongly orthogonal. Note that strongly orthogonal roots are also orthogonal for the scalar product in h∗

R.

An element H ∈ ˆh is regular if α(H) 6= 0 for all α ∈ R. Denote by C(R) the set of the Weyl chambers of R. They are the connected components of the set of regular elements of h_R. For C ∈ C(R), and H ∈ C, the set R+_{(C) = {α ∈ R | α(H) > 0} is}

independent of the choice of H ∈ C : it is called the set of positive roots with respect to C. The set R−(C) = R+_(Copp_{), for C}opp _{= {−H | H ∈ C}, is the complement}

of R+(C) in R and is called the set of negative roots with respect to C. A Weyl chamber C also defines a partial order relation ”≺C” in h∗_R, by :

(2.1) η ≺C ξ if η(A) < ξ(A) for all A ∈ C.

In particular R+_{(C) = {α ∈ R | α} C 0}.

With ˆgα = {X ∈ ˆg| [H, X] = α(H) X ∀H ∈ ˆh}, we set

(2.2) Q = {α ∈ R | ˆgα ⊂ q} .

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This is the parabolic set associated to q and ˆh. Parabolic sets of roots are abstractly defined by the two conditions :

α, β ∈ Q , α + β ∈ R =⇒ α + β ∈ Q (closedness) (2.4 i)

Q ∪ Qopp = R where Qopp = {−α | α ∈ Q} . (2.4 ii)

Given (2.4 i), condition (2.4 ii) is equivalent to the fact that Q ⊃ R+(C) for some Weyl chamber C ∈ C(R). We have

(2.5) q= qQ = ˆh⊕

X

α∈Q

ˆ gα

and the correspondence Q ←→ qQ is one-to-one between parabolic subsets of R

and parabolic subalgebras of ˆg containing ˆh. Given a parabolic set Q ⊂ R we set :

(2.6) Qr = Q∩Qopp{α ∈ Q | −α ∈ Q} and Qn _{= Q\Q}r _{= {α ∈ Q | −α /}_{∈ Q} .} Then (2.7) qn = X α∈Qn ˆ gα

is the nilradical of q, i.e. the set of the elements Z of its radical r(q) for which adˆg(Z) is nilpotent, and

(2.8) qr = ˆh⊕ X

α∈Qr ˆ gα

a reductive complement of qn in q. The complex parabolic Lie subalgebra q of ˆg is its own normalizer and the normalizer of its nilradical qn_:

(2.9) q= {Z ∈ ˆg| [Z, q] ⊂ q} = {Z ∈ ˆg| [Z, qn] ⊂ qn} . If A ∈ h_R, then the set

(2.10) QA = {α ∈ R | α(A) ≥ 0}

is parabolic, with Qr

A = {α ∈ R | α(A) = 0} and QnA = {α ∈ R | α(A) > 0}.

Vice versa, if Q is parabolic, set δ = P{α | α ∈ Qn_{}, and define T}

δ ∈ h_R by

(Tδ|A) = δ(A) for all A ∈ hR. Then Q = QTδ = {α ∈ R | (α|δ) ≥ 0}. The set of A ∈ h_R for which Q = QA is in fact a relatively open convex cone in h_R.

When Q = QA for some A ∈ hR, we shall also write qA for qQA.

The sets Qnassociated to parabolic Q’s are called horocyclic (see [War72, §1.1]). The correspondence Qn ←→ qn ₌P

α∈Qnˆgα is one-to-one between horocyclic sets of roots in R and nilradicals of complex parabolic Lie subalgebras containing ˆh.

Given a parabolic subset Q ⊂ R, we use the notation Q−nfor its opposite horo-cyclic set [Qn]opp = R \ Q : the corresponding nilpotent algebra q−n =P

α∈Q−nˆgα is a complement of q in ˆg.

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2.2. PARABOLIC SUBALGEBRAS AND COMPLEX FLAG MANIFOLDS 13

To a parabolic Q ⊂ R we associate the set of Weyl chambers :

(2.11) C(R, Q) = {C ∈ C(R) | R+(C) ⊂ Q} = {C ∈ C(R) | R+(C) ⊃ Qn} . We denote by B(C) the simple roots of R+(C), for C ∈ C(R). Every α ∈ R can be written in a unique way as a linear combination, with integral coefficients (either all ≥ 0 or all ≤ 0), of the simple roots in B(C):

(2.12) α = X

β∈B(C)

kβ_α(C)β .

We define the support of a root α as:

(2.13) supp_C(α) = {β ∈ B(C) | kβ_α(C) 6= 0} . If Q is parabolic, C ∈ C(R, Q), and ΦC(Q) = B(C) ∩ Qn, then

(2.14) Qn= {α ∈ R+(C) | supp_C(α) ∩ ΦC(Q) 6= ∅} . The correspondence (2.15) B(C) ⊃ ΦC ←→ q = ˆh ⊕ X α∈R+_(C) ˆ gα ⊕ X α∈R−(C) supp_C(α)∩ΦC=∅ ˆ gα

is one-to-one between subsets ΦC of B(C) and complex parabolic Lie subalgebra of

ˆ

g that contain ˆh and have an associated parabolic set Q with C ∈ C(R, Q).

Having fixed a Weyl chamber C ∈ C(R) and ΦC ⊂ B(C), we shall denote by

qΦC the complex parabolic Lie subalgebra of ˆg defined by the right hand side of (2.15) and by QΦC the corresponding parabolic set.

We denote by W(R) the Weyl group of R, (i.e. the group of isometries of h∗_R generated by the symmetries ξ −→ sα(ξ) = ξ − 2[(ξ|α)/kαk2] α for α ∈ R) and by

A(R) the group of all isometries of h∗_R (with respect to the scalar product defined above) that transform R into itself. For C ∈ C(R) we denote by AC(R) the

sub-group of A(R) consisting of the elements w ∈ A(R) for which w(R+(C)) = R+(C). Then A(R) = AC(R) o W(R).

We define W(R, Q) and A(R, Q) as the subgroups of W(R) and A(R), re-spectively, that transform Q into itself. Then we have Chevalley’s Lemma (see e.g. [War72, Theorem 1.1.2.8]) :

Lemma 2.1. The group W(R, Q) is generated by the symmetries sα with

α ∈ Qr. If C ∈ C(R, Q), then the symmetries sα with α ∈ B(C) \ ΦC(Q) generate

W(R, Q) and A(R, Q) is a semidirect product A(R, Q) = AC(R, Q) o W(R, Q),

with AC(R, Q) = AC(R) ∩ A(R, Q).

Let ˆG be a connected complex Lie group with Lie algebra ˆg. If Q is any Lie subgroup of ˆG with complex Lie subalgebra q that is parabolic in ˆg, then Q is closed, connected and coincides with its normalizer in ˆG and is the normalizer of its Lie algebra for the adjoint representation :

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The homogeneous space M = ˆG/Q is compact and simply connected. Since the center Z( ˆG) of ˆG is contained in all its parabolic subgroups, the choice of different connected complex Lie groups ˆG yields the same M. Hence we can consider the complex flag manifold M = ˆM (ˆg, q) as an object associated simply to the pair (ˆg, q). We also recall (see [Wol69, §2.7]) that the integral cohomology H∗_{(M, Z) is} torsion free and 0 in odd degrees.

If ˆh is a Cartan subalgebra of ˆg contained in q and Q ⊂ R(ˆg, ˆh) the parabolic set of roots associated to q, the complex dimension of M = ˆM (ˆg, q) equals the number of roots in Qn.

2.2 Parabolic CR algebras and CR manifolds

A CR algebra (g, q) is called parabolic if g is finite dimensional and q is a parabolic subalgebra of its complexification ˆg.

By the results stated above, if (g, q) is a parabolic CR algebra, then all the homogeneous spaces ˜M = ˜M (g, q), M = M (g, q), and M = ˆM (ˆg, q) are well de-fined. We recall that ˆG is the complex connected and simply connected Lie group with Lie algebra ˆg, the groups G and Q are the analytic subgroups of ˆG with Lie algebras g and q, respectively. Then M = ˆG/Q is a complex flag manifold and M is an orbit in M of the real form G of ˆG.

We say that M is a parabolic CR manifold.

Vice versa, if G is a connected real form of the complex semisimple Lie group ˆ

G, then all G-orbits in the complex flag manifolds M = ˆM (ˆg, q) are homogeneous CR manifolds of the form M = M (g, q0), for some parabolic complex Lie subalgebra q0 of ˆg, conjugated to q by an inner automorphism.

It is worth noticing that, in the definition of the homogeneous CR manifold M (g, q) = G/G+, we can define the isotropy G+ = G+(g, q) by

(2.17) G+ = {g ∈ G | Adˆg(g)(q) = q} .

Since the center of G is always contained in G+, we obtain an equivalent definition

of M (g, q) if we substitute to ˆG any connected complex Lie group ˆG0 with the same Lie algebra ˆg and to G the analytic subgroup G0 of ˆG0 with Lie algebra g. However, it is more convenient to fix a simply connected ˆG, since in this case, by [BT72, Corollaire 4.7], we have :

(2.18) G = ˆGσ = {g ∈ ˆG | σ(g) = g} ,

where σ : ˆG −→ ˆG is the anti-holomorphic involution of ˆG corresponding to the conjugation σ of ˆg defined by the real form g.

We begin by proving some general facts about parabolic CR algebras, and their associated CR manifolds.

Proposition 2.2. A parabolic CR algebra (g, q) is effective if and only if : (i) g is semisimple, (ii) no simple ideal of ˆg is contained in q ∩ q.

An effective parabolic CR algebra (g, q) with g simple is either totally complex or ideal nondegenerate.

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2.3. ADAPTED CARTAN SUBALGEBRAS AND CARTAN INVOLUTIONS 15

Proof. The statement follows by observing that: (a) for a parabolic (g, q) the radical r of g is contained in g+; (b) if an ideal a of ˆg is contained in q ∩ q, then

a+ a is the complexification of an ideal b of g contained in g+.

Proposition 2.3. Let (g, q) be an effective parabolic CR algebra and let g = g1 ⊕ · · · ⊕ g` be the decomposition of g into the direct sum of its simple

ideals. Then :

(i) q = q1⊕ · · · ⊕ q` where qj = q ∩ ˆgj for j = 1, . . . , `;

(ii) for each j = 1, . . . , `, (gj, qj) is an effective parabolic CR algebra;

(iii) (g, q) is ideal (resp. weakly, strictly) nondegenerate if and only if for each j = 1, . . . , `, the CR algebra (gj, qj) is ideal (resp. weakly, strictly)

nonde-generate;

(iv) (g, q) is fundamental if and only if for each j = 1, . . . , `, the CR algebra (gj, qj) is fundamental.

(v) We have (∼= meaning biholomorphic or CR equivalence) : ˆ

M (ˆg, q) ∼= ˆM (ˆg1, q1) × · · · × ˆM (ˆg`, q`) ,

˜

M (g, q) ∼= ˜M (g1, q1) × · · · × ˜M (g`, q`) ,

M (g, q) ∼= M (g1, q1) × · · · × M (g`, q`) .

Proof. In fact ˆg=L`j=1ˆgj is a decomposition of ˆginto a direct sum of ideals.

The decomposition (i) of q follows then from the decomposition ˆh=L`

j=1ˆh ∩ ˆgj

of any Cartan subalgebra of ˆg contained in q (see [Bou05, Ch.VII,§2,Prop.2]). The proof of the other statements is straightforward.

2.3 Adapted Cartan subalgebras and Cartan involutions

When q is parabolic in ˆg, its conjugate ¯q with respect to the real form g is also parabolic in ˆg. Therefore the intersection q ∩ ¯qcontains a Cartan subalgebra ˆhthat is invariant under conjugation. The intersection h = ˆh∩ g is a Cartan subalgebra of g, contained in g+ = q ∩ g.

A Cartan subalgebra h of g contained in g+ = q ∩ g is said to be adapted to

(g, q). We also have :

Proposition 2.4. Let (g, q) be an effective parabolic CR algebra, with isotropy subalgebra g+ = q ∩ g. The elements A of the radical r(g+) of g+ for

which adg(A) : g −→ g is nilpotent, form a nilpotent ideal n of g+. It admits a

reductive complement g0 in g+:

(2.19) g+ = n ⊕ g0.

The reductive subalgebra g0 is uniquely determined modulo inner automorphisms

of g+ from the subgroup generated by those of the form exp adg+(X) with X ∈ n. Proof. Indeed q, being parabolic, contains the semisimple and nilpotent parts of its elements. If X ∈ q belongs to the real form g, then also its semisimple and nilpotent parts belong to g. Therefore g+ is splittable, i.e. contains the semisimple

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and nilpotent part of its elements and we can apply [Bou05, Prop.7, §5, Ch.VII] to

obtain our statement.

Let z0 be the center and s0 = [g0, g0] the semisimple ideal of g0. Then

(2.20) g0 = z0⊕ s0.

Thus, a Cartan subalgebra h ⊂ g+ of g can be taken as the direct sum

(2.21) h = z0⊕ h0

of the center z0 of g0 and a Cartan subalgebra h0 of s0. Vice versa, every Cartan

subalgebra h of g adapted to (g, q) has the form (2.21) for some reductive subalgebra g0 of g+.

It is also convenient to consider a Cartan decomposition (see e.g. [Bou05]) :

(2.22) g= k ⊕ p

of g, corresponding to a Cartan involution ϑ . The set k = {X ∈ g | ϑ(X) = X} of fixed points of ϑ is a maximal compact Lie subalgebra of g and p = {X ∈ g | ϑ(X) = −X} its orthogonal for the Killing form κgof g. Any ϑ-invariant Cartan subalgebra

hof g decomposes into the direct sum h = h+_{⊕ h}− _{of its compact (or toroidal) part}

h+ = h ∩ k ⊂ k and its noncompact (or vector part) h− = h ∩ p ⊂ p.

We say that the Cartan decomposition (2.22) is adapted to the effective par-abolic CR algebra (g, q) if k contains a maximal compact Lie subalgebra of g+.

Then :

Lemma 2.5. If a Cartan decomposition (2.22) is adapted to the parabolic CR algebra (g, q), then every Cartan subalgebra h of g that is adapted to (g, q) is con-jugate, modulo an inner automorphism of g+, to a ϑ-invariant Cartan subalgebra

h0 of g that is adapted to (g, q).

Vice versa, if h is a Cartan subalgebra of g adapted to (g, q), then there exists a Cartan decomposition (2.22), adapted to (g, q), such that h = (h ∩ k) ⊕ (h ∩ p).

In particular, if {qi| i ∈ I} is a family of complex parabolic Lie subalgebras of

ˆ

gsuch thatT

i∈Iqi is parabolic in ˆg, then there exist both a Cartan decomposition

(2.22) and a Cartan subalgebra h of g, compatible with (2.22), that are adapted to

all the (g, qi)’s.

We say that (ϑ, h) is an adapted Cartan pair for (g, q) if :

(i) ϑ is the Cartan involution of a Cartan decomposition (2.22) adapted to (g, q) ; (ii) h is a ϑ-invariant Cartan subalgebra of g contained in g+ = g ∩ q.

Being σ : ˆg 3 X −→ ¯X ∈ ˆg the conjugation in ˆg associated to the real form g, and having fixed (2.22), we also consider the conjugation τ : ˆg −→ ˆg of ˆg with respect to its compact real form u = k ⊕ i p and use the same symbol ϑ to denote the C-linear extension to ˆg of the Cartan involution ϑ of g. We obtain in this way three commuting involutions σ, τ, ϑ of ˆg, each being the composition product of the other two :

(2.23) τ = ϑ ◦ σ = σ ◦ ϑ , σ = ϑ ◦ τ = τ ◦ ϑ , ϑ = σ ◦ τ = τ ◦ σ . In particular u is invariant under σ: σ(u) = u.

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2.5. THE FIBER OF A G-EQUIVARIANT FIBRATION 17

Let (ϑ, h) be a Cartan pair adapted to (g, q). Then h_R decomposes as : h_R = h− ⊕ i h+ _{and h}

R is a Cartan subalgebra of a split real form gR of ˆg. The

involutions σ, τ and ϑ transform h_R into itself. Hence, by transposition, they define involutions on h∗_R, that we still denote by the same symbols σ, τ and ϑ, and that transform the set of roots R into itself. We set ¯α = σ(α) for all α ∈ h∗_R. We have : (2.24) τ (α) = −α , ϑ(α) = − ¯α ∀α ∈ h∗_R.

The reductive complement qr of qn in q of (2.8) is q ∩ τ (q), while the reductive complement g0 of n in g+ of (2.19) can be taken equal to g+∩ ϑ(g+).

2.4 The fundamental and weakly nondegenerate reductions

We consider the CR fibrations of Proposition 1.5 in the special case of a parabolic CR algebra.

Theorem 2.6. Every effective parabolic CR algebra (g, q) admits a unique g-equivariant CR fibration (g, q) −→ (g, ˆg0), where ˆg0 ⊃ q is the complexification of a real parabolic subalgebra g0 of g, and the fiber is fundamental. The basis (g, ˆg0) is a totally real parabolic CR algebra and also the fiber (g0, q) is parabolic.

This yields a G-equivariant CR fibration π : M (g, q) −→ M (g, ˆg0) with compact basis. Each connected component of the fiber is CR diffeomorphic to M (g0, q), hence of finite type.

Proof. Let (g, q) be an effective parabolic CR algebra. The complex sub-algebra q00 generated by q + ¯q is parabolic in ˆg because contains q, and is the complexification of a real parabolic subalgebra g0 of g because ¯q00 = q00. Then (1.6) yields a g-equivariant CR fibration with a totally real basis. The fiber is (g0, q). This is parabolic because q, being parabolic in ˆg, is also parabolic in ˆg0 ⊂ ˆg.

The final statement follows from the commutative diagram : M (g, q) −−−−→ M (ˆˆ g, q) π   y ˆπ   y M (g, ˆg0) −−−−→ ˆM (ˆg, ˆg0)

that yields an embedding of each fiber of π : M (g, q) −→ M (g, ˆg0) into a fiber of ˆ

π : ˆM (ˆg, q) −→ ˆM (ˆg, ˆg0). The basis M (g, ˆg0) is compact because g ∩ ˆg0 is parabolic

in g.

Theorem 2.7. Let (g, q) be an effective parabolic CR algebra. Then there is a unique g-equivariant CR fibration (g, q) −→ (g, q0) with a weakly nondegenerate basis (g, q0) and a totally complex fiber. The basis (g, q0) is a parabolic CR algebra. Proof. We recall from [MN05, §5], that q0 is the unique maximal subalgebra of ˆg that contains q and is contained in q + q. Clearly q0 is parabolic because it

contains the parabolic subalgebra q.

2.5 The fiber of a G-equivariant fibration

Next we investigate the general structure of the fiber of a G-equivariant fibration M (g, q) −→ M (g, q0) for a pair of complex parabolic subalgebras q ⊂ q0 of ˆg.

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Theorem 2.8. Let q ⊂ q0 be complex parabolic Lie subalgebras of ˆg. With g0 = g ∩ q0, the CR algebra (g0, ¯q0 ∩ q) is the fiber of the g-equivariant fibration (g, q) −→ (g, q0_{) (see Chapter 1).}

The nilradical n0 of g0, consisting of the adg-nilpotent elements of the radical

r(g0) of g0, has a reductive complement g0₀ in g0 such that : (i) The CR algebra (g0₀, ˆg0₀∩ q) is parabolic.

(ii) The fiber (g0, ˆg0 ∩ q) is the semidirect sum of the parabolic CR algebra (g0₀, ˆg0₀∩ q) and of the nilpotent CR algebra (n0, ˆn0∩ q).

(iii) The nilpotent CR algebra (n0, ˆn0∩ q) is totally complex.

(iv) The connected components of the fibers of the G-equivariant fibration π : M (g, q) −→ M (g, q0_{) are CR diffeomorphic to the Cartesian product}

of a parabolic CR manifold M (g0₀, q ∩ ˆg0₀) and of a Euclidean complex manifold (∼_{= C}`_).

Proof. Fix a Cartan pair (ϑ, h), that is adapted for both (g, q) and (g, q0). Since q ⊂ q0 and q is parabolic, we have the inclusions : qr _{⊂ q}0 r _{and q}n _{⊃ q}0 n_.

The complexification of the fiber g0 is : ˆ g0 = q0∩ ¯q0 = ˆg0₀_{o ˆ}n0, where : _ˆ_g0 0 = q0 r∩ ¯q0 r, ˆ n0 = (q0 r∩ ¯q0 n) ⊕ (q0 n∩ ¯q0 r) ⊕ (q0 n∩ ¯q0 n) = (q0∩ ¯q0 n) + (q0 n∩ ¯q0) . Thus g0 = g0₀_{o n}0, where n0 = ˆn0∩ g is a real form of the nilradical ˆn0 of q0∩ ¯q0 and g0₀ := ˆg0₀∩ g a reductive complement of n0 in g0. We have : ˆ g0∩ q = (ˆg0₀_{o ˆ}n0) ∩ q = (ˆg0₀_{∩ q) o (ˆn}0∩ q) , so that : (g0, ˆg0∩ q) = (g0₀, ˆg0₀_{∩ q) o (n}0, ˆn0∩ q) .

The complex Lie subalgebra ˆg0₀∩ q is parabolic in ˆg0₀, because ˆg0₀ is reductive, q is parabolic in ˆg, and ˆg0₀∩ q contains a Cartan subalgebra of ˆg0₀ and of ˆg.

Note that ˆn0∩ q is contained in, but in general not equal to, the nilradical ˆn of q∩ ¯q. We have : ˆ n0∩ q ⊃ q0 n∩ ¯q0, so that : ˆ n0∩ q + ˆn0_{∩ q ⊃ q}0 n_{∩ ¯}_q0 _{+ q}0 _{∩ ¯}_q0 n _{= ˆ}_n0

shows that actually :

ˆ

n0∩ q + ˆn0_{∩ q = ˆ}_n0

and the nilpotent CR algebra (n0, ˆn0∩ q) is totally complex.

The G-equivariant fibration π : M (g, q) −→ M (g, q0_{) is the restriction of the ˆ}_G

equivariant fibration ˆπ : ˆM (ˆg, q) −→ ˆM (ˆg, q0). The typical fiber F of ˆπ is Q0/Q. Since q is a parabolic complex Lie subalgebra of q0, the fiber F is a complex flag manifold and, in particular, is compact, connected and simply connected. Thus the typical fiber F of π : M (g, q) −→ M (g, q0_{) is a submanifold of a complex flag}

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Denote still by τ : ˆG −→ ˆG the involution of ˆG associated to the conjugation τ : ˆg −→ ˆg with respect to the compact real form u = k ⊕ i p. Since q0n ⊂ qn_,

the fiber F can be viewed also as a flag manifold of the reductive complex closed connected Lie subgroup Q0 r = Q0 ∩ τ (Q0_{) of ˆ}_{G. The fiber F is contained in the}

orbit F0 ⊂ F of the closed complex Lie subgroup ˆG0 := Q0∩σ(Q0) of Q0. The group

ˆ

G0 is connected because it contains a Cartan subgroup of ˆG, and decomposes into the semidirect product

(2.25) Gˆ0 = ˆG0₀_oNˆ0,

where ˆG0₀ and ˆN0 are the analytic complex Lie subgroups of ˆG generated by the Lie subalgebras ˆg0₀ and ˆn0, respectively. We have ˆG0₀ = Q0 r ∩ σ(Q0 r_{), so that}

ˆ

G0₀ is closed in ˆG. Moreover, since adˆg(Z) is nilpotent for all Z ∈ ˆn, by Engel’s

theorem and the semisimplicity of ˆg, we obtain that exp : ˆn0 −→ ˆN0 is an analytic diffeomorphism, and ˆN0 is Euclidean.

The validity of (iv) is then a consequence of the next Proposition.

Proposition 2.9. Let ˆN0 be a connected nilpotent complex Lie group with complex Lie algebra ˆn0 and n0 a real form of ˆn0. Let N0 be the real analytic Lie subgroup of ˆN0 with Lie algebra n0, and Q0 a closed connected complex Lie

sub-group of ˆN0, with Lie algebra q0 ⊂ ˆn0, and set N = Q0∩ N0. Assume that the CR

algebra (n0, q0) is totally complex. With E = N0/N and ˆE = ˆN0/Q0, the natural

map E −→ ˆE obtained from the inclusion N0 ,→ ˆN0 is a diffeomorphism.

Proof. The condition that (n0, q0) is totally complex is equivalent to the

equal-ity n0+q0 = ˆn0. Since ˆn0 is nilpotent, this equality implies (see the proof below) that

the map N0× Q0 3 (n, q) −→ n · q ∈ ˆN0 is onto, and hence the inclusion N0 ,→ ˆN0

yields, by passing to the quotients, a smooth one-to-one map f : E −→ ˆE. We note that E = N0/N is a complex manifold with the homogeneous CR structure defined by the CR algebra (n0, q0). With this complex structure on E, and with

the complex structure that ˆE inherits from ˆN0, the map f is holomorphic. Being one-to-one, f is a biholomorphism.

We give here a simple argument to prove that ˆN0 = N0Q0.

Consider the lower central series ˆ

n= C(0)(ˆn0) ⊃ C(1)(ˆn0) = [ˆn0, ˆn0] ⊃ · · · ⊃ C(h)(ˆn0) =C(h−1)(ˆn0), ˆn0 ⊃ · · · · · · ⊃ C(m)_(ˆ_n0_{) =}_C(m−1)_(ˆ_n0_{), ˆ}_n0_{= {0} .}

Since ˆn0 is nilpotent and ˆN0 is connected, the exponential map exp : ˆn0 −→ ˆN0 is surjective. Let Z ∈ ˆn0. We want to prove that there is g ∈ N0 such that g−1· exp(Z) ∈ Q0. To this aim, let X ∈ n0 and W ∈ q0 be such that Z = X + W .

Let Z1 ∈ ˆn0 be such that exp(Z1) = exp(−X) exp(Z) exp(−W ). We claim that, if

Z ∈ Ch(ˆn0), then Z1 ∈ Ch+1(ˆn0).

While proving this claim, we can assume that ˆN0 is also simply connected, so that all Lie subgroups ˆN0_h = exp C(h)(ˆn0) are closed and simply connected. For

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each integer h ≥ 0 we have a commutative diagram : ˆ n0 −−−−→exp Nˆ0 πh   y   y ph ˆ n0/Ch+1(ˆn0) −−−−→ ˆ[exp] N0/ ˆN0_h+1

where [exp] denotes the exponential map on the quotient. If Z ∈ Ch(ˆn0), then πh(Z)

belongs to the center of the quotient Lie algebra ˆn0/Ch+1(ˆn0). Hence we obtain : [exp] (πh(Z1))= [exp](−πh(X)) · [exp](πh(Z)) · [exp](πh(X − Z))

= [exp](−πh(X))) · [exp] (πh(Z) + πh(X − Z))

= [exp](−πh(X))) · [exp](πh(X))) = 1_Nˆ0_{/ ˆ}_N0 h+1. Since [exp] is a diffeomorphism, we obtain that Z ∈ Ch+1(ˆn0).

We show by recurrence that for every Z ∈ C(m−i)(ˆn0) there is some X ∈ n such that exp(−X) · exp(Z) ∈ Q0. This is trivially true when m = 0, as Z = 0 in this

case. If Z ∈ C(m−i)(ˆn0) for some i > 0, and X ∈ n0 is such that X − Z ∈ q0, then

exp(−X) · exp(Z) · exp(X − Z) = exp(Z1) for some Z1 ∈ C(m−i+1)(ˆn0). By the

recursive assumption, there is X1 ∈ n0 such that exp(−X1) exp(Z1) ∈ Q0. Then

g = exp(X1) · exp(X) ∈ N0 and g−1 · exp(Z) ∈ Q0. For i = m we obtain our

contention.

From Theorem 2.8 we obtain :

Theorem 2.10. Let M = M (g, q) and M0 = M (g, q0) be parabolic CR mani-folds. If q ⊂ q0 ⊂ q +¯q, then the G-equivariant fibration M −→ M0 is a CR fibration and has a totally complex simply connected fiber.

Proof. We already noted in Chapter 1 that the CR algebra associated to the fiber F of the fibration M −→ M0, and hence F itself, is totally complex when q ⊂ q0 _{⊂ q + ¯}_{q. By Theorem 2.8, the connected components of the fiber are the}

product of a Euclidean complex nilmanifold and of a manifold M (g0₀, ˆg0₀∩ q), for a totally complex parabolic CR algebra (g0₀, ˆg0₀ ∩ q). This M (g0

0, ˆg00 ∩ q) is an open

orbit of a connected real form G0₀ of a connected complex Lie group ˆG0₀ with Lie algebra ˆg0₀, and thus is simply connected by [Wol69, Theorem 5.4]. Corollary 2.11. Let (g, q0) be the weakly nondegenerate reduction of the effective parabolic CR algebra (g, q). Then

(2.26) f : M = M (g, q) −→ M0 = M (g, q0)

is a G-equivariant CR fibration with complex simply connected fibers. We give here a simple general criterion that ensures the existence and the con-nectedness of the fiber of some G-equivariant fibrations.

Proposition 2.12. Keep the notation introduced above. The isotropy sub-group G+ is the closed real semi-algebraic subgroup of G :

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The isotropy subgroup G+ admits a Chevalley decomposition

G+ = G0o N

where :

(i) N is a unipotent, closed, connected, and simply connected subgroup with Lie algebra n;

(ii) G0 is a reductive Lie subgroup, with Lie algebra g0, and is the centralizer

of its center z = zg0(g0) in G :

G0 = ZG(z) = {g ∈ G | Adg(g)(H) = H ∀H ∈ z} .

Proof. Let g ∈ G+. Then Adg(g)(g0) is a reductive complement of n in g+.

Since all reductive complements of n are conjugated by an inner automorphism from Adg+(N), we can find a gn ∈ N such that Adg+(g

−1

n g)(g0) = g0. Consider

the element gr = g−1n g. We have :

Adg(gr)(g0) = g0, Adˆg(gr)(qΦ) = qΦ,

Adˆg(gr)(qnΦ) = qnΦ, Adˆg(gr)(¯qΦ) = ¯qΦ,

because gr∈ Q ∩ ¯Q. We consider the parabolic subalgebra of ˆg defined by :

qΦ0 = qn_Φ⊕ (qr_Φ∩ ¯q_Φ) = qn_Φ+ (q_Φ∩ ¯q_Φ) .

It has the property that qr_Φ0 = ¯qr_Φ0 is the complexification of g0. Clearly

Adˆg(gr)(qΦ0) = q_Φ0 and Ad_ˆ_g(g_r)(qr

Φ0) = qr_Φ0. Hence gr ∈ QrΦ0 and the state-ment follows because Qr_Φ0 = Z_Gˆ(zqr

Φ0(q r

Φ0)) is the centralizer of the center of its Lie algebra and z is a real form of zqr

Φ0(q r

Φ0).

Proposition 2.13. Let (g, q), (g, q0) be two effective parabolic CR alge-bras. Assume that g+ = q ∩ g ⊂ g0+ = q0 ∩ g and that g+ contains a

Car-tan subalgebra h that is maximally noncompact among the CarCar-tan subalgebras of g that are contained in g0₊. Then the germ of local G-equivariant submersion (M (g, q), o) −→ (M (g, q0), o0), defined by the projection g/g+ −→ g/g0, extends to a

G-equivariant fibration π : M (g, q) −→ M (g, q0_{) with connected fibers.}

Proof. Decompose G+ = G0o N and G0+ = G00 o N0. Let H = ZG(h) =

{h ∈ G | Adg(h)(H) = H , ∀H ∈ h} be the Cartan subgroup of G corresponding to

h. We have Adˆg(h)(q) = q and Adˆg(h)(q0) = q0 for all h ∈ H. Hence H ⊂ G0∩ G00.

Since h is maximally noncompact in g0₀ and a fortiori in g0, by [Kna02, Prop.7.90],

all connected components of G0₀ and G0, and also of G0+ and G+, intersect H.

The connected component of the identity G0₊ of G+ is contained in the connected

component [G0₊]0 of the identity in G0₊. Since G+ is generated by G0+ and H,

and likewise G0₊ is generated by [G0₊]0 _{and H, we obtain at the same time that}

G+ ⊂ G0+ and that the fiber G0+/G+ is connected.

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Theorem 2.14. Let (g, q) and (g, q0) be two effective parabolic CR algebras such that:

q∩ ¯q= q0∩ ¯q0.

Then the CR manifolds M = M (g, q) and M0 = M (g, q0) are diffeomorphic, by a G-equivariant diffeomorphism.

Proof. Let h be a Cartan subalgebra of g, contained in g+ = q∩g = g0+ = q0∩g

and maximally noncompact as a Cartan subalgebra of g+.

Let A, A0 ∈ h_R be such that q = qA, q0 = qA0. We can assume that q 6= q0, so that A and A0 are linearly independent. Then we set At = A + t(A0 − A),

for 0 ≤ t ≤ 1, so that A0 = A and A1 = A0. Let us take a partition

t0 = 0 < t1 < · · · < tm−1 < tm = 1 such that the rank of adgˆ(At) is constant for t in

the open intervals tj−1 < t < tj, 1 ≤ j ≤ m, so that qAt = qAt0 for tj−1 < t, t 0 _{< t}

j.

Let Mj = M (g, qA_tj), for 0 ≤ j ≤ m, and Nj = M (g, qA_{(tj−1+tj )/2}), for 1 ≤ j ≤ m.

Since : qA_{(tj−1+tj )/2} ⊂ qA_tj−1 ∩ qA_tj, there are G-equivariant maps :

Mj−1 fj

←−−−− Nj Fj

−−−−→ Mj

for all 1 ≤ j ≤ m. By Proposition 2.13, all these maps, being covering maps with connected fibers, are diffeomorphisms. Thus :

(Fm◦ fm−1) ◦ (Fm−1◦ fm−1−1 ) ◦ · · · ◦ (F1◦ f1−1) : M −→ M 0

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CHAPTER 3

Fit Weyl chambers and CR geometry of M (g, q)

Let M (g, q) be a parabolic CR manifold and (ϑ, h) a Cartan pair adapted to (g, q). In this Chapter we introduce some special Weyl chambers, that we call S-fit and V-fit, and describe some geometric properties of M , namely fundamentality and weak nondegeneracy, in terms of properties of the simple roots associated to these special Weyl chambers.

We keep the notation of the preceding chapters, for roots, parabolic sets, Car-tan decomposition, etc. In particular, we denote by σ : h∗_R 3 α −→ ¯α ∈ h∗_R the adjoint map of the restriction to h_R = h−⊕ i h+ _{of the conjugation in ˆ}_g _{defined by}

the real form g. We say that a root α is real if ¯α = α, imaginary if ¯α = −α, complex if ¯α 6= ±α and denote by Rre, Rimand Rcp the sets of real, imaginary and complex

roots, respectively. When α is imaginary, the eigenspace ˆgα is contained either in ˆ_k _{= C ⊗}

Rk, or in ˆp = C ⊗R p. In the first case we say that α is compact, in the

second that α is noncompact. Thus Rim is the disjoint union of the set R• of the

compact and of the set R∗ of the noncompact imaginary roots : Rim= R•∪ R∗.

3.1 S-fit and V-fit Weyl chambers

The conjugation σ defines an involution in h∗_R that belongs to the group A(R) of isometries of the root system R. Vice versa, every involution σ in A(R) can be obtained from a conjugation with respect to a real form g of ˆg. Note that, in general, σ does not uniquely determine the isomorphism class of g. Let us describe the structure of an arbitrary involution σ in A(R) :

Theorem 3.1. Let σ be an involution in A(R). Then there exist: a set of pairwise strongly orthogonal roots α1, . . . , αm in R, with σ(αj) = −αj for

j = 1, . . . , m, a Weyl chamber C ∈ C(R), and an involution  ∈ AC(R), with

(αi) = αi, and hence commuting with sαi, for all i = 1, . . . , m, such that : (3.1) σ =  ◦ sα1 ◦ · · · ◦ sαm;

(3.2) α ∈ R+(C) =⇒ either σ(α) = −α or σ(α) ∈ R+(C)

Recall that two roots α, β ∈ R are strongly orthogonal if α ± β /∈ R.

Proof. Let F−(σ) = {α ∈ h∗_R| σ(α) = −α}, take a maximal subset α1, . . . , αm

of pairwise orthogonal roots in F−(σ)∩R and consider  = σ◦sα1◦· · ·◦sαm. We have (αi) = σ(−αi) = αi for all i = 1, . . . , m. We claim that (α) 6= −α for all α ∈ R.

Indeed, if there was α ∈ R with (α) = −α, from (α|αi) = ((α)|(αi)) = −(α|αi)

we obtain that (α|αi) = 0 for all i = 1, . . . , m. Hence sαi(α) = α for all α and 23

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therefore σ(α) = (α) = −α, contradicting the fact that α1, . . . , αm was a maximal

system of pairwise orthogonal roots in R ∩ F−(σ).

To obtain that α1, . . . , αm are strongly orthogonal it suffices to choose the

sequence α1, . . . , αm with a maximal sum P m i=1kαik

2_{. Indeed, if α}

j and αh are

or-thogonal, but not strongly oror-thogonal, then both αj+αhand αj−αh are roots.

Set-ting α0_i = αi for i 6= j, h, and α0j = αj+αh, α0h = αj−αh, we obtain a new sequence

α0₁, . . . , α_m0 of pairwise orthogonal roots in F−(σ) ∩ R. It is contained in a maximal one and the inequalityPm

i=1kα0ik

2 ₌ Pm

i=1kαik2 + kαjk2+ kαhk2 >P m

i=1kαik2,

contradicts the maximality ofPm

i=1kαik 2_.

We claim that there exists a Weyl chamber C such that :

(∗) (R+(C)) = R+(C) .

Indeed (∗) is equivalent to (B(C)) ⊂ R+_{(C). For a Weyl chamber C, denote}

by nC the number of the elements in R+(C) ∩ (R+(C)). Fix C with nC

max-imum. If C does not satisfy (∗), take α ∈ B(C) with (α) /∈ R+_{(C) and}

con-sider the chamber C0 = sα(C). From R+(C0) = (R+(C) \ {α}) ∪ {−α} and

(−α) ∈ R+_{(C) \ {α} ⊂ R}+_(C0_{), we obtain n}

C0 = n_C + 1, contradicting our choice of C. Hence C satisfies (∗) and therefore also (3.2). This completes the

proof.

Using Theorem 3.1, we obtain the formula : (3.3) σ(β) = (β) −

m

X

j=1

β|α∨_j αj, with α∨j = 2αj/kαjk2, ∀β ∈ h∗_R.

Likewise, we have the following :

Theorem 3.2. Let σ be an involution in A(R). Then there exists a set of pair-wise strongly orthogonal roots δ1, . . . , δm ∈ R, with σ(δj) = δj for j = 1, . . . , m,

a Weyl chamber C ∈ C(R) and an involution $, that commutes with sδj, satisfies $(δj) = −δj for all j = 1, . . . , m, and transforms C into Copp, such that :

(3.4) σ = $ ◦ sδ1 ◦ · · · ◦ sδm,

(3.5) α ∈ R+(C) =⇒ either σ(α) = α or σ(α) ∈ R−(C) .

Proof. We take σ0 = s0◦ σ, where s0 is the symmetry with respect to the

origin of h∗_R. By the preceding Theorem, σ0 =  ◦ sδ1◦ · · · ◦ sδm, where  ∈ AC(R) for some C ∈ C(R), and δ1, . . . , δm is a maximal system of strongly orthogonal roots

in F−(σ0) ∩ R = {α ∈ R | σ(α) = α}, with (δj) = δj. The statement follows by

taking $ = s0◦ .

With $ and δ1, . . . , δm as in Theorem 3.2, we obtain the formula :

(3.6) σ(β) = $(β) +

m

X

j=1

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3.1. S-FIT AND V-FIT WEYL CHAMBERS 25

A Weyl chamber C ∈ C(R) that satisfies condition (3.2) (resp. (3.5)) is said to be2 S-adapted (resp. V-adapted) to the conjugation σ.

For general CR algebras (g, q), there could be no adapted Cartan subalgebras h that are either maximally compact or maximally noncompact in g . This is a major drawback in the classification of the orbits of G in M (see e.g. the references in [BL02]), but, while discussing fundamentality, weak nondegeneracy and some topological properties, it turns out that the choice of h is not as crucial as that of special Weyl chambers C in C(R, Q). In general C(R, Q) may not contain any Weyl chamber that is either S- or V-adapted to σ. In the following lemmas we describe chambers in C(R, Q) that are as close as possible to being S- or V-adapted.

Lemma 3.3. Let (ϑ, h) be a Cartan pair adapted to (g, q). Then there exists a Weyl chamber C ∈ C(R, Q) that satisfies the equivalent conditions :

(i) If α /∈ Rim, α C 0, and ¯α ≺C 0, then both α and − ¯α belong to Qn.

(ii) ¯α C 0 for all α ∈ B(C) \ (ΦC ∪ Rim) .

Assume that C satisfies the equivalent conditions (i) and (ii). Then :

(iii) If moreover h is maximally noncompact among the Cartan subalgebras of g contained in g+ = q ∩ g, then B(C) ∩ R∗ ⊂ ΦC.

Proof. Choose C ∈ C(R, Q) with a maximal R+(C) ∩ σ (R+(C)). Then (ii) is satisfied. Indeed, if there was α ∈ B(C) \ (ΦC ∪ Rim) with ¯α ≺C 0, we would take

C0 = sα(C). Then R+(C0) = (R+(C) \ {α}) ∪ {−α} ⊂ Q, so that C0 ∈ C(R, Q),

and R+_(C0_{) ∩ σ (R}+_(C0_{)) % R}+_{(C) ∩ σ (R}+_{(C)), yielding a contradiction. Clearly}

(i) ⇒ (ii). Vice versa, if α = P

β∈B(C)k β

αβ ∈ R+(C) and ¯α ≺C 0 , then either

α ∈ Rim, or else there is some β ∈ suppC(α) ∩ Rcp with ¯β ≺C 0 ; by (ii), we have

β ∈ ΦC and hence α ∈ Qn. The same argument, applied to − ¯α, shows that also

− ¯α ∈ Qn. This completes the proof of the equivalence (i) ⇔ (ii).

Finally, if α ∈ (B(C) ∩ R∗) \ ΦC, both α and ¯α = −α belong to Q. Let

Γ = {(Xα, Hα)α∈R} be a Chevalley system, as in [Bou05]. Then Tα = Xα− X−α∈

p∩g+ (for this construction cf. [Sug59]) is a semisimple element of g that commutes

with all elements of h− and of j+= {H ∈ h+| α(H) = 0}. Hence j = h−_{⊕ R T} α⊕ j+

is a Cartan subalgebra of g, contained in g+, with j− = h−_{⊕ R T}α % h−. Thus, if

h− is maximal, we have B(C0) ∩ R∗ ⊂ ΦC.

An alternative construction of a Weyl chamber C satisfying (i) and (ii) of Lemma 3.3 is the following (which is a particular case of a general construction that will be described in Chapter 4). Fix a Weyl chamber C0 ∈ C(R) that is

S-adapted to σ (recall that this means σ (R+_(C

0) \ Rim) ⊂ R+(C0)), and consider

the Borel subalgebra :

bC0 = ˆh⊕ M α∈R+_(C 0) ˆ gα ⊂ ˆg. Then b = qn _{⊕ (q}r_{∩ b}

C0) is a Borel subalgebra of ˆg, corresponding to a Weyl chamber C ∈ C(R, Q) that satisfies (i) and (ii) of Lemma 3.3.

2_{If we choose h maximally noncompact, then in an S-adapted Weyl chamber C the}

conjuga-tion can be described by a Satake diagram; if instead we take h with a maximal compact part, in a V-adapted Weyl chamber the conjugation is described by a Vogan diagram (see e.g. [Ara62], [Kna02].

Orbits of real forms in complex flag manifolds

Tesi di Dottorato